Today, I called a cab to get me and my cat home from the vet. After a 40 minute wait and three calls to the taxi company, the operator told me that the driver she sent to my location just confirmed that he already has a woman with my name and a cat in his car. FML

If OP only gave her first name and it was something quite common, it doesn't seem that extraordinary that there could have been other women with cats and the same name leaving the vet.
Or, another likely scenario: another woman leaves the vet with her cat, she sees a taxi there. She opens the door, driver asks if this is [OP's name], the other woman claims that she is OP, she gets OP's taxi.

I know exactly what happened. you were making this call from the past. or maybe the cab was really a time travel machine. or maybe you and your cat were cloned and sent into a parallel universe inwhich you were late in the past but on time in the future...*Gasp*

I think what maybe happened was the OP called for a cab, the cab got there an asked a random woman (who was either waiting outside or just walking out with her cat) "are you [OP's name here]?" And maybe the woman lied and took the cab in hopes OP's card was already on file to pay.
Or maybe I've been reading too much "Not Always Right".

Maybe they don't have Uber or Lyft in Germany.
I feel your pain, OP. Before my boyfriend and I started using Uber and Lyft, trying to get a cab home from the vet was pretty tricky. Even though she stays in her carrier, more than one cabbie tried to charge us the pet fee.